A cusped prismatic shell-like body under the action of concentrated moments

The elastic equilibrium problem of a cusped prismatic shell-like body, when its projection is a half-plane x ₂ ≥ 0, under the action of a concentrated moment is solved in the explicit form within the framework of the zero approximation of I.Vekua’s hierarchical models of prismatic shells. The thickness of the prismatic shell-like body is proportional to the coordinate x ₂ raised to a non-negative exponent. When the exponent equals to zero, the above solution contains the well-known solution of the classical Carothers’ problem [1] in the case of an elastic half-plane (see also [2], §39).     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

A numerical method for the Benjamin-Ono equation

This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu _t +uu _x −Hu _xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.     

Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape

A general method is presented for the rigorous solution of Eshelby’s problem concerned with an arbitrary shaped inclusion embedded within one of two dissimilar elastic half-planes in plane elasticity. The bonding between the half-planes is considered to be imperfect with the assumption that the interface imperfections are uniform. Using analytic continuation, the basic boundary value problem is reduced to a set of two coupled nonhomogeneous first-order differential equations for two analytic functions defined in the lower half-plane which is free of the thermal inclusion. Using diagonalization, the two coupled differential equations are decoupled into two independent nonhomogeneous first-order differential equations for two newly defined analytic functions. The resulting closed-form solutions are given in terms of the constant imperfect interface parameters and the auxiliary function constructed from the conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. The method is illustrated using several examples of an imperfect interface. In particular, when the same degree of imperfection is realized in both the normal and tangential directions between the two half-planes, a thermal inclusion of arbitrary shape in the upper half-plane does not cause any mean stress to develop in the lower half-plane. Alternatively, when the imperfect interface parameters are not equal, then a nonzero mean stress will be induced in the lower half-plane by the thermal inclusion of arbitrary shape in the upper half-plane. Detailed results are presented for the mean stress and the interfacial normal and shear stresses caused by a circular and elliptical thermal inclusion, respectively. Results from these calculations reveal that the imperfect bonding condition has a significant effect on the internal stress field induced within the inclusion as well as on the interfacial normal and shear stresses existing between the two half-planes especially when the inclusion is near the imperfect interface.     

Ann-dimensional search problem with restricted questions

The problem is the following: How many questions are necessary in the worst case to determine whether a pointX in then-dimensional Euclidean spaceR ⁿ belongs to then-dimensional unit cubeQ ⁿ, where we are allowed to ask which halfspaces of (n−1)-dimensional hyperplanes contain the pointX? It is known that ⌌3n/2⌍ questions are sufficient. We prove here thatcn questions are necessary, wherec≈1.2938 is the solution of the equationx log₂ x−(x−1) log₂ (x−1)=1.     

On disjointly representable sets

A system of setsE ₁,E ₂, ...,E _k ⊂X is said to be disjointly representable if there existx ₁,x ₂, ...,x _k teX such thatx _i teE _j ⇔i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems. Dedicated to Paul Erdős on his seventieth birthday     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

On a biharmonic equation involving nearly critical exponent

This paper is concerned with a biharmonic equation under the Navier boundary condition , u > 0 in Ω and u = Δu = 0 on ∂Ω, where Ω is a smooth bounded domain in , n ≥ 5, and ε > 0. We study the asymptotic behavior of solutions of (P _−ε) which are minimizing for the Sobolev quotient as ε goes to zero. We show that such solutions concentrate around a point x ₀ ∈Ω as ε → 0, moreover x ₀ is a critical point of the Robin’s function. Conversely, we show that for any nondegenerate critical point x ₀ of the Robin’s function, there exist solutions of (P _−ε) concentrating around x ₀ as ε → 0. Finally we prove that, in contrast with what happened in the subcritical equation (P _−ε), the supercritical problem (P _+ε) has no solutions which concentrate around a point of Ω as ε → 0. Work finished when the authors were visiting Mathematics Department of the University of Roma “La Sapienza”. They would like to thank the Mathematics Department for its warm hospitality. The authors also thank Professors Massimo Grossi and Filomena Pacella for their constant support.     

A proximal point algorithm for minimax problems

This paper presents a proximal point algorithm for solving discretel _∞ approximation problems of the form minimize ∥Ax−b∥_∞. Let ε_∞ be a preassigned positive constant and let ε _l ,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε _l < ε_∞. Then, starting from an arbitrary pointz ₀, the proposed method generates a sequence of points z _l ,l= 0,1,2,..., via the rule . One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within a finite number of iterations. The smaller are the numbers ε _l the smaller is the number of iterations. In fact, if ε ₀ is sufficiently small then z₁ solves the original minimax problem. The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax problem of the form minimize where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize , and ify solves the dual thenx=A ^T y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate the feasibility of our ideas.     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

Uniform convergence of wavelet solution to the sideways heat equation

We consider the problem uxx（x, t） = ut（x, t）, 0 ≤ x 〈 1, t ≥ 0, where the Cauchy data g（t） is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g（t） can produce a big alteration on its solution （if it exists）. We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space Vj. In the previous papers, the theoretical results concerning the error estimate are L2-norm and the solutions aren＇t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].     

Effective transverse elastic moduli of composites at non-dilute concentration of a random field of aligned fibers

We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov–Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c². Received: March 4, 2003; revised: August 8, 2003     

Applications of the Differential Calculus to Nonlinear Elliptic Operators with Discontinuous Coefficients

The paper concerns Dirichlet’s problem for second order quasilinear non-divergence form elliptic equations with discontinuous coefficients. We start with suitable structure, growth, and regularity conditions ensuring solvability of the problem under consideration. Fixing then a solution u ₀ such that the linearized at u ₀ problem is non-degenerate, we apply the Implicit Function Theorem. As a result we get that for all small perturbations of the coefficients there exists exactly one solution u ≈ u ₀ which depends smoothly (in W ^2,p with p larger than the space dimension) on the data. For that, no structure and growth conditions are needed and the perturbations of the coefficients can be general L ^∞-functions of the space variable x. Moreover, we show that the Newton Iteration Procedure can be applied in order to obtain a sequence of approximate (in W ^2,p ) solutions for u ₀.     

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x ₁…,x _n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f ⁻¹{0} inf ⁻¹{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x ₁,x ₂]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

Weakly Singular Integral Operators in Weighted L∞–Spaces

We study integral operators on (−1, 1) with kernels k(x, t) which may have weak singularities in (x, t) with x ∈N₁, t ∈N₂, or x=t, where N₁,N₂ are sets of measure zero. It is shown that such operators map weighted L^∞–spaces into certain weighted spaces of smooth functions, where the degree of smoothness is the higher the smoother the kernel k(x, t) as a function in x is. The spaces of smooth function are generalizations of the Ditzian-Totik spaces which are defined in terms of the errors of best weighted uniform approximation by algebraic polynomials.     

On trigonometric polynomials least deviating from zero

We give a solution of the problem about trigonometric polynomials with a given leading harmonic and least deviating from zero in measure; more precisely, with respect to the functional μ(f _n ) = mes {t ∈ [0, 2π]: |f _n (t)| ≥ 1}. We give a solution of a related problem about the minimal value over compact sets (from the real line) of a given measure of least uniform deviation from zero on a compact set for trigonometric polynomials with a fixed leading harmonic. Published in Russian in Doklady Akademii Nauk, 2009, Vol. 425, No. 6, pp. 733–736. Presented by Academician A.M. Il’in November 13, 2008 The article was translated by the authors.     

Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow

We discuss the problem of non-linear oscillations of a clamped thermoelastic plate in a subsonic gas flow. The dynamics of the plate is described by von Kármán system in the presence of thermal effects. No mechanical damping is assumed. To describe the influence of the gas flow we apply the linearized theory of potential flows. Our main result states that each weak solution of the problem considered tends to the set of the stationary points of the problem. A similar problem was considered in [27], but with rotational inertia accounted for, i.e. with the additional term −αΔu_tt,α > 0, and the same result on stabilization was obtained. There was introduced the decomposition of the solution such that the one term tends to zero and the other is compact in special (“local energy”) topology. This decomposition enables us to prove the main result. But the case of rotational inertia neglected (α = 0) appears more difficult. Low a priori smoothness of u_t in the case α = 0 prevents us to construct such a decomposition. In order to prove additional smoothness of u_t we use analyticity of the corresponding thermoelastic semigroup proved in [25]. The isothermal variant of this problem with additional mechanical damping term −εΔu_t , ε > 0 was considered in [13] and stabilization to the set of stationary solutions to the problem was proved. The problem, considered in the present work can also be regarded as an extension of the result of [18] to the case when gas occupies an unbounded domain.     

A non-trivial explicit solution for the 14-velocity Cabannes kinetic model

We study the shock wave problem for the Cabannes 14-velocity model of the Boltzmann equation in one space dimension (x$x$-axis) and obtain a non-trivial explicit solution which asymptotically connects two particular Maxwellian states. We prove that such a solution exists provided that a suitable condition on the differential elastic cross sections hold.     

A 2D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity

We consider a system of hyperbolic integro-differential equations of SH waves in a visco-elastic porous medium. In this work, it is assumed that the visco-elastic porous medium has weakly horizontally inhomogeneity. The direct problem is the initial-boundary problem: the initial data is equal to zero, and the Neumann-type boundary condition is specified at the half-plane boundary and is an impulse function. As additional information, the oscillation mode of the half-plane line is given. It is assumed that the unknown kernel has the form K(x,t)=K₀(t)+ϵxK₁(t)+…, where ϵ is a small parameter. In this work, we construct a method for finding K₀,K₁ up to a correction of the order of O(ϵ²).